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Theorem nfrexya 2428
 Description: Not-free for restricted existential quantification where 𝑦 and 𝐴 are distinct. See nfrexxy 2426 for a version with 𝑥 and 𝑦 distinct instead. (Contributed by Jim Kingdon, 3-Jun-2018.)
Hypotheses
Ref Expression
nfralya.1 𝑥𝐴
nfralya.2 𝑥𝜑
Assertion
Ref Expression
nfrexya 𝑥𝑦𝐴 𝜑
Distinct variable group:   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem nfrexya
StepHypRef Expression
1 nftru 1407 . . 3 𝑦
2 nfralya.1 . . . 4 𝑥𝐴
32a1i 9 . . 3 (⊤ → 𝑥𝐴)
4 nfralya.2 . . . 4 𝑥𝜑
54a1i 9 . . 3 (⊤ → Ⅎ𝑥𝜑)
61, 3, 5nfrexdya 2424 . 2 (⊤ → Ⅎ𝑥𝑦𝐴 𝜑)
76mptru 1305 1 𝑥𝑦𝐴 𝜑
 Colors of variables: wff set class Syntax hints:  ⊤wtru 1297  Ⅎwnf 1401  Ⅎwnfc 2222  ∃wrex 2371 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077 This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-cleq 2088  df-clel 2091  df-nfc 2224  df-rex 2376 This theorem is referenced by:  nfiunya  3780  nffrec  6199  nfsup  6767  caucvgsrlemgt1  7437  nfsum1  10899  zsupcllemstep  11368  bezout  11427
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